| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2015 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Derive equation from integral condition |
| Difficulty | Standard +0.3 This question involves straightforward integration of exponentials, algebraic manipulation to rearrange the equation, and applying a given iterative formula. The integration is routine (antiderivatives of e^(3x) and e^x), the rearrangement requires basic algebra with exponentials and logarithms, and the iteration is mechanical computation. While it combines multiple techniques, each step is standard A-level material with no novel insight required, making it slightly easier than average. |
| Spec | 1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Integrate to obtain \(e^{3x}+5e^x\) | B1 | |
| Apply both limits and subtract for expression of form \(k_1e^{3x}+k_2e^x\) | M1 | |
| Obtain \(e^{3a}+5e^a=106\) or similarly simplified equivalent | A1 | |
| Rearrange and introduce logarithms | M1 | |
| Confirm given answer \(a=\frac{1}{3}\ln(106-5e^a)\) | A1 | [5] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Use the iterative formula correctly at least once | M1 | |
| Obtain final answer \(1.477\) | A1 | |
| Show sufficient iterations to justify accuracy to 3 d.p. or show sign change in interval \((1.4765, 1.4775)\) | A1 | [3] |
## Question 5:
### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Integrate to obtain $e^{3x}+5e^x$ | B1 | |
| Apply both limits and subtract for expression of form $k_1e^{3x}+k_2e^x$ | M1 | |
| Obtain $e^{3a}+5e^a=106$ or similarly simplified equivalent | A1 | |
| Rearrange and introduce logarithms | M1 | |
| Confirm given answer $a=\frac{1}{3}\ln(106-5e^a)$ | A1 | [5] |
### Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Use the iterative formula correctly at least once | M1 | |
| Obtain final answer $1.477$ | A1 | |
| Show sufficient iterations to justify accuracy to 3 d.p. or show sign change in interval $(1.4765, 1.4775)$ | A1 | [3] |
---5 It is given that $\int _ { 0 } ^ { a } \left( 3 \mathrm { e } ^ { 3 x } + 5 \mathrm { e } ^ { x } \right) \mathrm { d } x = 100$, where $a$ is a positive constant.\\
(i) Show that $a = \frac { 1 } { 3 } \ln \left( 106 - 5 \mathrm { e } ^ { a } \right)$.\\
(ii) Use an iterative formula based on the equation in part (i) to find the value of $a$ correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
\hfill \mbox{\textit{CAIE P2 2015 Q5 [8]}}